# How do you simplify (p^3-1)/(5-10p+5p^2)?

Jan 24, 2017

$\frac{{p}^{3} - 1}{5 - 10 p + 5 {p}^{2}} = \frac{{p}^{2} + p + 1}{5 \left(p - 1\right)}$ or $\frac{{p}^{2} + p + 1}{5 p - 5}$

#### Explanation:

To simplify $\frac{{p}^{3} - 1}{5 - 10 p + 5 {p}^{2}}$, we should first factorize numerator and denomiantor.

${p}^{3} - 1 = {p}^{3} - {p}^{2} + {p}^{2} - p + p - 1$

= ${p}^{2} \left(p - 1\right) + p \left(p - 1\right) + 1 \left(p - 1\right)$

= $\left({p}^{2} + p + 1\right) \left(p - 1\right)$

and $5 - 10 + 5 {p}^{2} = 5 \left({p}^{2} - 2 p + 1\right)$

= $5 \left({p}^{2} - p - p + 1\right) = 5 \left(p \left(p - 1\right) - 1 \left(p - 1\right)\right) = 5 \left(p - 1\right) \left(p - 1\right)$

Hence $\frac{{p}^{3} - 1}{5 - 10 p + 5 {p}^{2}}$

= $\frac{\left({p}^{2} + p + 1\right) \left(p - 1\right)}{5 \left(p - 1\right) \left(p - 1\right)}$

= $\frac{\left({p}^{2} + p + 1\right) \cancel{\left(p - 1\right)}}{5 \left(p - 1\right) \cancel{\left(p - 1\right)}}$

= $\frac{{p}^{2} + p + 1}{5 \left(p - 1\right)}$ or $\frac{{p}^{2} + p + 1}{5 p - 5}$